Faster Minimization of Total Weighted Completion Time on Parallel Machines

TL;DR

This paper introduces a faster algorithm for minimizing total weighted completion time on parallel identical machines, improving upon the classic Lawler and Moore algorithm for certain problem parameter ranges.

Contribution

The authors develop a novel algorithm that outperforms the traditional pseudo-polynomial time algorithm in specific cases, marking a significant advancement in scheduling problem solutions.

Findings

New algorithm is faster for small parameter values

Achieves improved runtime over Lawler and Moore in certain scenarios

First such improvement in over 50 years

Abstract

We study the classical problem of minimizing the total weighted completion time on a fixed set of $m$ identical machines working in parallel, the $Pm||\sum w_jC_j$ problem in the standard three field notation for scheduling problems. This problem is well known to be NP-hard, but only in the ordinary sense, and appears as one of the fundamental problems in any scheduling textbook. In particular, the problem served as a proof of concept for applying pseudo-polynomial time algorithms and approximation schemes to scheduling problems. The fastest known pseudo-polynomial time algorithm for $Pm||\sum w_jC_j$ is the famous Lawler and Moore algorithm from the late 1960's which runs in $\tilde{O}(P^{m-1}n)$ time, where $P$ is the total processing time of all jobs in the input. After more than 50 years, we are the first to present an algorithm, alternative to that of Lawler and Moore, which is faster for certain range of the problem parameters (e.g., when their values are all $O(1)$).